Complex symmetric Toeplitz operators on the generalized derivative Hardy space.

The generalized derivative Hardy space S α , β 2 (D) consists of all functions whose derivatives are in the Hardy and Bergman spaces as follows: for positive integers α, β, S α , β 2 (D) = { f ∈ H (D) : ∥ f ∥ S α , β 2 2 = ∥ f ∥ H 2 2 + α + β α β ∥ f ′ ∥ A 2 2 + 1 α β ∥ f ′ ∥ H 2 2 < ∞ } , where H (D) denotes the space of all functions analytic on the open unit disk D . In this paper, we study characterizations for Toeplitz operators to be complex symmetric on the generalized derivative Hardy space S α , β 2 (D) with respect to some conjugations C ξ , C μ , λ . Moreover, for any conjugation C, we consider the necessary and sufficient conditions for complex symmetric Toeplitz operators with the symbol φ of the form φ (z) = ∑ n = 1 ∞ φ ˆ (− n) ‾ z ‾ n + ∑ n = 0 ∞ φ ˆ (n) z n . Next, we also study complex symmetric Toeplitz operators with non-harmonic symbols on the generalized derivative Hardy space S α , β 2 (D) . [ABSTRACT FROM AUTHOR]